Problem 55 - Thomas Calculus

Question

Solve the initial value problems in Exercises \(53-58\). $$ \frac{d s}{d t}=8 \sin ^{2}\left(t+\frac{\pi}{12}\right), \quad s(0)=8 $$

Step-by-Step Solution

Answer

The solution is \( s(t) = 4t - 2 \sin\left(2t + \frac{\pi}{6}\right) + 9 \).

Step 1: Recognize the Differential Equation

We are given the differential equation \( \frac{d s}{d t} = 8 \sin^2\left(t + \frac{\pi}{12}\right) \) with the initial condition \( s(0) = 8 \). The goal is to find the solution \( s(t) \) that satisfies both the differential equation and the initial condition.

Step 2: Integrate the Right-Hand Side

To solve for \( s(t) \), we need to integrate the right-hand side of the differential equation. The integral becomes: \[ s(t) = \int 8 \sin^2\left(t + \frac{\pi}{12}\right) \, dt. \]

Step 3: Use Trigonometric Identity

We can make integration easier by using the trigonometric identity \( \sin^2(x) = \frac{1 - \cos(2x)}{2} \). Apply it to get: \[ 8 \sin^2\left(t + \frac{\pi}{12}\right) = 4 - 4 \cos\left(2t + \frac{\pi}{6}\right). \]

Step 4: Integrate Each Term Separately

Integrate each term separately: \[ \int (4 - 4 \cos\left(2t + \frac{\pi}{6}\right)) \, dt = \int 4 \, dt - \int 4 \cos\left(2t + \frac{\pi}{6}\right) \, dt. \] The integrals are \( 4t \) and \( -2 \sin(2t + \frac{\pi}{6}) \), respectively.

Step 5: Write the General Solution

Combine the results to write the general solution: \[ s(t) = 4t - 2 \sin\left(2t + \frac{\pi}{6}\right) + C, \] where \( C \) is the integration constant.

Step 6: Apply the Initial Condition

Use the initial condition \( s(0) = 8 \) to solve for \( C \). Substitute \( t = 0 \) into the equation: \[ 8 = 4(0) - 2 \sin\left(\frac{\pi}{6}\right) + C, \] which simplifies to \( 8 = -1 + C \). Thus, \( C = 9 \).

Step 7: Write the Final Solution

Substitute \( C = 9 \) back into the general solution to get the specific solution: \[ s(t) = 4t - 2 \sin\left(2t + \frac{\pi}{6}\right) + 9. \]

Key Concepts

Initial Value Problems

Initial value problems are a subset of differential equations where the solution aims to satisfy an initial condition. In these problems, we try to find a function that not only satisfies the differential equation but also passes through a specific point known as the initial condition. For example, in the problem at hand, our differential equation is given as \(\frac{d s}{d t}=8 \sin^{2}\left(t+\frac{\pi}{12}\right)\), and the initial condition is \(s(0) = 8\).
The initial condition helps us determine the constant of integration that emerges from solving the differential equation, thus providing a unique solution. Without an initial condition, we'd only find a general solution with an arbitrary constant. Initial value problems are commonly encountered in physics and engineering, where they describe systems that evolve from a known starting state.

Trigonometric Identities

Trigonometric identities are key tools in simplifying complex trigonometric expressions, especially when dealing with integration. In this exercise, to integrate \(8 \sin^2\left(t + \frac{\pi}{12}\right)\), we apply the trigonometric identity \(\sin^2(x) = \frac{1 - \cos(2x)}{2}\).
This identity reduces the complexity of the sine squared function, making it easier to integrate. By recognizing and applying the correct identities, we can transform difficult integrals into more manageable forms. This skill is crucial for solving many types of mathematical problems, particularly those involving trigonometric functions.

Integration

Integration is the mathematical process of finding the integral of a function, often used to determine accumulated quantities, such as area under a curve. In this initial value problem, the integration is performed on the function \(8 \sin^2\left(t + \frac{\pi}{12}\right)\) after simplifying it using the trigonometric identity.
This is done by splitting it into separate terms: \(\int (4 - 4 \cos\left(2t + \frac{\pi}{6}\right)) \, dt = \int 4 \, dt - \int 4 \cos\left(2t + \frac{\pi}{6}\right) \, dt\).
The integration of constant terms is straightforward, while integrating functions like cosine involves recognizing derivative patterns or using substitution if needed. Integration is a core concept in calculus used frequently to solve differential equations, like our problem here.

Separable Equations

Separable equations are a category of differential equations that can be manipulated algebraically such that all terms involving one variable appear on one side of the equation, and the terms involving the other variable are on the opposite side.
In this exercise, we implicitly deal with a separable equation, as the equation \(\frac{d s}{d t}=8 \sin^2\left(t + \frac{\pi}{12}\right)\) can be solved by direct integration once formulated appropriately.
Separable equations are among the simplest types of differential equations to solve and are a standard topic in any differential equations course. Mastering separable equations is a key step in understanding and solving more complex differential equations effectively.